Chapter 8.5, Problem 2E

Using Table G, find the critical value(s) for each. Show the critical and noncritical regions, and state the appropriate null and alternative hypotheses. Use σ2 = 225.

a. α = 0.01, n = 17, right-tailed

b. α = 0.025, n = 20, left-tailed

c. α = 0.01, n = 13, two-tailed

d. α = 0.025, n = 29, left-tailed

To determine

To find: The critical value for right tailed test.

Answer to Problem 2E

The critical value for right tailed test is 32.000.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2>225$

Explanation of Solution

The level of significance is $\alpha =0.01$, the sample size is 17 and the test is right-tailed.

Answer:

Degrees of freedom:

$\begin{array}{lll}df\hfill & =\hfill & n-1\hfill \\ \hfill & =\hfill & 17-1\hfill \\ \hfill & =\hfill & 16\hfill \end{array}$

From “Table G: The chi square distribution”, the critical values for the $a$ value in the table of 0.01 with 16 degrees of freedom is 32.000.

Software Procedure:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =17-1\\ =16\end{array}$

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.01 with 16 degrees of freedom is 32.000.

Software Procedure:

Step-by-step procedure to obtain the critical region using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 16.
• Click the Shaded Area tab.
• Choose Probability value and Right Tail for the region of the curve to shade.
• Enter the Probability value as 0.01.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2>225$ 

To determine

To find: The critical value for left tailed test.

Answer to Problem 2E

The critical value for left tailed test is 8.907.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2<225$

Explanation of Solution

The level of significance is $\alpha =0.025$, the sample size is 20 and the test is left-tailed.

Solution:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =20-1\\ =19\end{array}$

If the test is left tail, the level of significance is subtracted from 1. That is, $1-0.025=0.975$.

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.975 with 19 degrees of freedom is 8.907.

Software Procedure:

Step-by-step procedure to obtain the critical region using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 19.
• Click the Shaded Area tab.
• Choose Probability value and Left Tail for the region of the curve to shade.
• Enter the Probability value as 0.025.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2<225$

To determine

To find: The critical values for two tailed test.

Answer to Problem 2E

The critical values for two tailed test is 3.074 and 28.299, respectively.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2\ne 225$ 

Explanation of Solution

Given info:

The level of significance is $\alpha =0.01$, the sample size is 13 and the test is two-tailed.

Solution:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =13-1\\ =12\end{array}$

The area to the right of larger value is, $\frac{0.01}{2}=0.005$ and the area to right of the smaller value is $1-\frac{0.01}{2}=0.995$.

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.005 and 0.995 with 12 degrees of freedom is 28.299 and 3.074, respectively.

Software Procedure:

Step-by-step procedure to obtain the critical region using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 12.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.01.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2\ne 225$

To determine

To find: The critical value for left tailed test.

Answer to Problem 2E

The critical value for left tailed test is 15.308.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2<225$

Explanation of Solution

Given info:

The level of significance is $\alpha =0.025$, the sample size is 29 and the test is left-tailed.

Solution:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =29-1\\ =28\end{array}$

If the test is left tail, the level of significance is subtracted from 1. That is, $1-0.025=0.975$.

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.975 with 28 degrees of freedom is 15.308.

Software Procedure:

Step-by-step procedure to obtain the critical region using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 28.
• Click the Shaded Area tab.
• Choose Probability value and Left Tail for the region of the curve to shade.
• Enter the Probability value as 0.025.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2<225$